A company owns two dealerships, both of which sell cars and trucks. Dealership A sells a total of 164 cars and trucks. Dealership B sells twice as many cars and half as many trucks as Dealership A, and sells a total of 229 cars and trucks. (4 points) a) An equation for the total cars and trucks for Dealership A: x + y = 229 b) An equation for the total cars and trucks for Dealership B: c) How many cars did Dealership A sell? d) Considering that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?
b) Let's define x as the number of cars sold by Dealership A, and y as the number of trucks sold by Dealership A. Equations for Dealership A and Dealership B can be written as:
Dealership A: x + y = 164
Dealership B: 2x + (1/2)y = 229
c) From equation a), we know that x + y = 164. Rearranging the equation to solve for x, we get:
x = 164 - y
Substituting this expression for x into equation b), we get:
2(164 - y) + (1/2)y = 229
Simplifying the equation:
328 - 2y + (1/2)y = 229
328 - 2y + y/2 = 229
Subtracting 328 from both sides:
-2y + y/2 = -99
Multiplying both sides by 2 to get rid of the fraction:
-4y + y = -198
Simplifying the equation:
-3y = -198
Solving for y:
y = (-198)/(-3) = 66
d) Substituting the value of y into equation a), we can solve for x:
x + 66 = 164
Subtracting 66 from both sides:
x = 164 - 66 = 98
So, Dealership A sold 98 cars, and Dealership B sold 66 trucks.
b) An equation for the total cars and trucks for Dealership B:
Let C be the number of cars sold by Dealership A.
Let T be the number of trucks sold by Dealership A.
Dealership A sells a total of 164 cars and trucks: C + T = 164
Dealership B sells twice as many cars and half as many trucks as Dealership A, and sells a total of 229 cars and trucks: 2C + (1/2)T = 229
c) How many cars did Dealership A sell?
From equation (a), we know that C + T = 164.
However, we need to calculate the value of C (number of cars sold by Dealership A).
We can subtract T from both sides of the equation to isolate C: C = 164 - T.
d) Considering that Dealership B sold half as many trucks as Dealership A, how many trucks did Dealership B sell?
From equation (b), we know that 2C + (1/2)T = 229.
Since we have the expression for C, we can substitute it into equation (b): 2(164-T) + (1/2)T = 229
Simplifying, we get: 328 - 2T + (1/2)T = 229
Combining like terms, we get: -3/2T = -99
Multiply both sides by -2/3 to solve for T: T = (-2/3)(-99) = 66
Therefore, Dealership A sold 164 - 66 = 98 cars, and Dealership B sold 66 trucks.
a) The total cars and trucks sold by Dealership A can be represented by the equation x + y = 164, where x represents the number of cars and y represents the number of trucks.
b) Dealership B sells twice as many cars and half as many trucks as Dealership A. So, if we let c represent the number of cars and t represent the number of trucks sold by Dealership B, the equation for the total cars and trucks for Dealership B would be c + t = 229.
c) To find out how many cars Dealership A sold, we can substitute the values into the equation for Dealership A: x + y = 164.
Since we only have one equation with two variables, we need more information to determine the values of x and y.
d) Dealership B sold half as many trucks as Dealership A. Let's assume the number of trucks sold by Dealership A is y. Therefore, Dealership B would have sold half of y, which can be represented as (1/2)y.